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Demand forecasting with high dimensional data: the case of SKU retail sales forecasting with intra- and inter-category

promotional information

Shaohui Maa,1

Robert Fildesb

Tao Huangc

a School of Economics and Management, Jiangsu University of Science and Technology,

China, 212003

b Lancaster Centre for Forecasting, Lancaster University, UK, LA1 4YX

c Kent Business School, University of Kent, UK, ME4 4AG

1 Corresponding author at: School of Economics and Management, Jiangsu University of Science and Technology,

ZhenJiang,212003, China. Tel.: +86 138 15179032. E-mail address: msh@tju.edu.cn (Shaohui Ma); r.fildes@lancaster.ac.uk (R. Fildes); t.huang@kent.ac.uk (T. Huang).

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Abstract

In marketing analytics applications in OR, the modeler often faces the problem of

selecting key variables from a large number of possibilities. For example, SKU level retail

store sales are affected by inter and intra category effects which potentially need to be

considered when deciding on promotional strategy and producing operational forecasts. But

no research has yet put this well accepted concept into forecasting practice: an obvious

obstacle is the ultra-high dimensionality of the variable space. This paper develops a four

steps methodological framework to overcome the problem. It is illustrated by investigating

the value of both intra- and inter-category SKU level promotional information in improving

forecast accuracy. The method consists of the identification of potentially influential

categories, the building of the explanatory variable space, variable selection and model

estimation by a multistage LASSO regression, and the use of a rolling scheme to generate

forecasts. The success of this new method for dealing with high dimensionality is

demonstrated by improvements in forecasting accuracy compared to alternative methods of

simplifying the variable space. The empirical results show that models integrating more

information perform significantly better than the baseline model when using the proposed

methodology framework. In general, we can improve the forecasting accuracy by 12.6

percent over the model using only the SKU’s own predictors. But of the improvements

achieved, 95 percent of it comes from the intra-category information, and only 5 percent from

the inter-category information. The substantive marketing results also have implications for

promotional category management.

Keywords: Analytics; OR in marketing; Forecasting; Retailing; Promotions

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1. Introduction

Many marketing problems require the analyst to understand the interactions of a large

number of potentially inter-related variables. For example, grocery retailers rely heavily on

accurate sales forecasts at SKU level when making business decisions in a wide range of

areas including marketing, production, inventory, and finance etc. Sales and promotional

effects in any one SKU are potentially affected by marketing and sales activities in a large

number of other categories – in other words, there are intra and inter-category variables that

may affect the target variable(s). However, identifying important variables from such a large

set of possibilities poses a serious modelling challenge- it is the subject of this paper.

In a retail forecasting system, product sales history, intra-category promotional schedules,

and inter-category promotional schedules are all potential rich sources of information which

may influence forecasting accuracy. When building product sales forecasting models, a series

of related but fundamental questions must be answered: which sources of information should

be inputted into the forecasting model? To what extent do different sources of information

contribute to forecasting accuracy improvements? And critically, how to manipulate the high

dimensional information to generate better forecasts?

The main challenge to be faced is that the dimensionality of promotional explanatory

variables grows very rapidly when cross-product promotional information is considered,

potentially much larger than the length of SKU sales history. The model may be easily

over-fitted or even cannot be estimated. To build a forecasting model for a SKU, when

considering both intra- and inter-category promotional interactions, the number of candidate

explanation variables is usually in the order of tens of thousands. With high dimensionality,

important predictors can be highly correlated with some unimportant ones, and the maximum

spurious correlation also grows with dimensionality (Fan and Lv, 2008).

Traditional methods which deal with the problem of high dimensionality include the

subset selection method, the penalized L-1 likelihood method, and the information summary

approach. The subset selection method and the penalized L-1 likelihood method both try, but

using different mechanisms, to find out the most influential variables affecting the dependent

variable. However, in the retail context, store managers may promote similar products

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simultaneously (e.g. different SKUs under the same brand), which makes the price and

promotional variables of different SKUs highly correlated to each other. As a result, these two

methods may select some unimportant predictors which are highly correlated with the

important predictors but fail to select the truly important predictors. The information

summary approach condenses the information of the vast number of variables (which we

cannot directly use due to high dimensionality) into a small number of factors but at the cost

of (potentially high) information loss.

To overcome the problem, a four step methodological framework is proposed in this

paper which consists of the identification of potentially influential categories, the building of

explanatory variable space, variable selection and model estimation by a multistage LASSO

(Least Absolute Shrinkage and Selection Operator) regression, followed by a scheme to

generate forecasts. The method breaks down the process of variables selection into three

stages: 1) to select variables related to promotional history of the focal product; 2) to select

intra-category variables and 3) finally, to select inter-category variables.

The development of a successful modelling system, necessarily automatic in order to deal

with the large number of SKUs, would also allow retailers to simulate the expected results

based on different promotional plans so that they can then optimize their promotional

schedules (Levy et al. 2004, Zhang et al. 2008). The need for an effective modelling and

forecasting system is therefore transparent. Existing studies in the literature have overlooked

the inter-categorical variables because available methods are incapable of effectively

integrating the useful information contained by these variables, as we discuss in the next

section.

In this paper, we focus on developing an automatic modelling approach which we

validate by applying it to the problem of forecasting many thousands of retail SKUs in order

to produce improved short term forecasts. Through a series of empirical data experiments, we

show that the proposed method of variable selection is an effective approach to simplifying

the dimensionality of the promotional marketing space: it improves forecasting accuracy

significantly by simplifying and integrating more retail information. But generally, the

inter-category information contributes limited accuracy improvements comparing to that of

intra-category information.

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The outline of the paper is as follows. In section two, we review existing related studies

and address their limitations. In Section three we discuss methodological issues. Section four

describes the data, introduces the experimental design and forecasting accuracy measures,

and presents the empirical results. Section five discusses the findings, offering conclusions as

to forecasting practice and further academic research.

2. Literature review

2.1 Model building with a large number of explanatory variables

We are in the era of massive automatic data collection, systematically obtaining many

measurements, not knowing which ones will be relevant to the phenomenon of interest

(Donoho, 2000). Traditional statistical methodology, assuming many observations and a few

well-chosen variables, are not designed to cope with this kind of explosive growth of

dimensionality of the observation vector. The increasing availability of data is thus creating

new challenges for the market modeller. There are, essentially, three different approaches to

address this problem. The first approach is concerned with finding the most influential subset

of predictors; the second approach builds predictive models based on summaries of the

predictor variables and the third approach is penalized (L-1) likelihood method which

automatically selects influential variables via continuous shrinkage.

Best subset selection is a popular class of the dimension reduction methods concerned

with finding the most influential subset of predictors in predictive modeling from a much

larger set of potential predictors. The best subset problem belongs to the class of NP-hard

problems known as induction of minimal structures (John et al., 1994). When the number of

potential predictors is large, the selection process cannot be solved exactly within an

acceptable amount of computation time. Consequently, heuristic optimization algorithms

have evolved, including iterative improvement algorithms (e.g., stepwise regression, forward

and backward feature selection algorithms) and stochastic search methods (e.g., Genetic

algorithms (Melab et al., 2002), simulated annealing (Meiri and Zahavi 2006)), to solve

larger scale combinatorial problems. However, the expensive computational cost still makes

best subset selection procedures infeasible for high-dimensional data analysis.

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The information summary approach to forecasting with high dimensional data is based

on the assumption that the relevant information is captured by a small number of factors

common to the predictor variables. A popular technique that combines the potentially relevant

predictors into new predictors is principal components. For example, basing the forecast

model on data summaries in the form of principal components, as in Stock and Watson

(2002), allows information from all the predictors to enter into the forecasts. Stock and

Watson (1999), Stock, Watson, and Marcellino (2003), and Forni et al. (2000, 2003), among

others, all find that diffusion factors based forecasts have smaller mean-squared errors than

forecasts based upon simple autoregressions and more elaborate structural models. A

criticism of factor augmented regressions is that the factors are estimated without taking into

account the dependent variable. Thus, when only a few factors are retained to represent the

variations of whole explanatory variable space, they might not have any predictive power for

the dependent variable whereas the discarded factors might be useful (Stock and Watson,

2002).

Penalized L-1 likelihood methods have been successfully developed over the last decades

to cope with high dimensionality. Penalized L-1 regression is called LASSO by Tibshirani

(1996) in the ordinary regression setting which has received much attention due to its

convexity and encouraging sparsity solutions. It minimizes the usual sum of squared errors,

with a bound on the sum of the absolute values of the coefficients. There has been much work

in recent years, applying and generalizing the LASSO and L1-like penalties to a variety of

problems (Tibshirani 2011). Efron et al. (2004) propose a fast and efficient least angle

regression (LARS) algorithm for variable selection, a simple modification of which produces

the entire LASSO solution path. A linear combination of L-1 and L-2 penalties is called an

elastic net by Zou and Hastie (2005), which encourages some grouping effects. Zou (2006)

introduced an adaptive lasso in a finite parameter setting.

L-1 type regularization does not eliminate the conflict between consistent model

selection and prediction. With high dimensionality, important predictors can be highly

correlated with some unimportant ones, and the maximum spurious correlation also grows

with dimensionality (Fan and Lv, 2008). But LASSO tends to arbitrarily select only one

variable among a group of predictors with high pairwise correlations. This may results in

some unimportant predictors that are highly correlated with the important predictors being

selected by LASSO while important predictors are missed.

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2.2 Intra- and inter-category promotional effects

The idea that demand in one product category can be affected by marketing efforts in

another is not new. In economics, products are considered complements (substitutes) if

lowering (raising) the price of one product leads to an increase in sales of another (Nicholson,

1998). Product substitutability and complementarity have long been natural ways to perceive

inter-category relationships.

Within one category, products of different brands, even the same brand in different

flavors or different pack sizes are usually regarded as substitutes for each other. A large body

of research supports the view that brands within a product category are substitutes for one

another (Frank and Massy, 1967; Kumar and Leone, 1988; Moriarty, 1985; Mulhern and

Leone, 1991; Walters, 1988, 1991). Researchers have found that the majority of the

promotional response stems from brand switching, the percentage of own-brand sales

elasticity with respect to a particular promotion that is due to brand-switching elasticity is

about 75%-84% (Gupta, 1988; Chiang, 1991; Chintagunta, 1993; Bucklin et al., 1998).

For many categories, consumer purchasing patterns are also affected by stimulating

purchases of nonpromoted complements to the promoted products (Berman and Evans, 1989;

Walters, 1988). For example, the promotion of a pie filling may stimulate sales of full-margin

pie shells, or the promotion of taco shells may increase sales of nonpromoted taco sauce. In

such cases, one promotion can increase the sales of products in two different categories. In

addition, inter- rather than intra-product substitution may also be the predominant influence

in certain product groups. Walters (1991), using store level SKU sales data, tested a

conceptual framework for retail promotion effects that includes brand substitution effects,

inter-store sales displacements and the purchase of complementary goods. He selected four

product categories in his study (spaghetti, spaghetti sauce, cake mix and cake frosting) and

found that both the complementary effects of promotion and the substitution effects of

promotion on brand sales are significant. Bandyopadhyay (2009) proposed a dynamic model

based on vector autoregression (VAR), and empirically studied intra- and inter-category

promotional effects with four brands of ice cream, two brands of topping, and three brands of

frozen yogurt. He found that a multiple-category model that includes brands from substitute

and complementary categories returns more accurate sales forecasts than does a

single-category model that includes brands from only a single category. Hruschka (2013)

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analyzed multi-category buying decisions of households by a finite mixture of multivariate

Tobit-2 models. He found 18% of all pairwise category correlations are significant. Studies

also showed that the cross-category impact of national brands on store brands appears to be

substantially greater than that of store brands on national brands (Wedel and Zhang 2004).

This means that the promotional effects are asymmetrical not only within but also across

categories.

Though existing research has provided evidence that promotions of one product can

influence the sales of another because of both intra- and inter-category effects, most of the

existing literature has focused on developing explanatory models, using a set of ad hoc

assumed product relationships to test the significance of the cross brand/category promotional

affects. Whether these theoretical findings can be applied in a real forecasting system to help

retailers improve the decision accuracy at SKU level is the question we concern ourselves

with in this research. This is a very different problem than those only concerned with

explanation and hypothesis testing. When we build forecasting models for tens thousands of

SKUs in a store, a problem size many retailers face, most of these existing theoretical models

lose their feasibility. For example, in a VAR model, the number of free parameters increases

quadratically with the number of variables in a system, and for even moderately-sized

systems the model becomes highly overparameterized relative to the number of available

observations. Even basic least square regression will not be applicable because the

dimensionality of cross category promotion explanatory variables is potentially much larger

than the sample size. In practice, we also cannot easily identify which product

complements/substitutes another. For example, beer and carbonated beverages could be either

substitutive or complementary, for people could drink them at different times in a day. And

even if we can specify a group of product categories within which possible promotional

interactive effects exist (no matter whether complementary or substitutive), we still cannot

easily specify which products in these categories interact with each other.

2.3 SKU sales forecasting

The basic SKU sales methods are univariate forecasting models which are based on time

series techniques that analyze past sales history in order to extract a demand pattern that is

projected into the future (Raju, 1995; Ord and Fildes, 2013). The techniques range from the

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simpler moving averages and exponential smoothing family to the more complicated Box–

Jenkins ARIMA approach, or the Exponential smoothing state space class of model

(Hyndman et al. 2002; Taylor, 2007). The methods do not take external factors such as price

changes and promotions into account (Alon, Qi & Sadowski, 2001). Gür Ali et al. (2009)

found that the simple time series techniques perform well for periods without promotions.

However, for periods with promotions, models with more inputs improve accuracy

substantially. Therefore, univariate forecasting methods are usually adopted as a benchmark

model in many studies (Gür Ali et al., 2009; Huang et al., 2014).

In order to improve SKU sales forecasting in the presence of promotions, many studies

have integrated the focal product’s promotional variables into their forecasting models. In

practice, many retailers use a base-times-lift approach to forecast product sales at the SKU

level (Cooper et al., 1999; Huang et al., 2014). The approach is a two-step procedure which

initially generates a baseline forecast from a simple time series models and then makes

adjustments for any incoming promotional events. The adjustments are estimated based on

the lift effect of the most recent price reduction and/or promotion, and also the judgements

made by brand managers (Fildes et al., 2008; Fildes et al., 2009). These judgmental

adjustments are common in practice, expensive and potentially prone to systematic errors

(Fildes & Goodwin, 2007; Fildes et al., 2008). Studies have shown that statistical models

usually performed better than the expert adjustments. (Trapero et al. 2013, Trapero et al.

2014). In the recent literature, some studies focus on how to make the adjustment more

effectively (Lee et al., 2007; Nikolopoulos, 2010), while others discuss how to integrate

statistical forecasts and managers’ judgment (Fildes et al., 2009, Trapero et al. 2011). Another

stream of studies uses a model-based forecasting system to forecast product sales by directly

taking into account the promotional information. These methods are usually based on

multiple regression models or data mining technologies whose exogenous inputs correspond

to the focus product’s own promotion features (Rinne and Geurts, 1988; Preston and Mercer,

1990; Cooper et al., 1999; Kuo, 2001; Aburto and Weber, 2007; Gür Ali et al., 2009, Gür Ali,

2013). For example, in Cooper et al. (1999), a promotion-event forecasting system called

PromoCast is reported, which uses a static cross-sectional regression analysis of SKU-store

sales under a variety of promotion conditions, with store and chain specific historical

performance information. The limitation of these studies is they overlook the potential

importance of price reductions and promotions of other influential products, nor do they

include time series dynamics.

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Forecasting product sales integrating influential products’ promotional information has

also been explored by previous researchers. A well know example is the SCAN*pro model

and its extensions which decompose sales for a brand into own- and cross -brand effects of

price, feature advertising, aisle displays, week effects, and store effects (Wittink et al., 1988;

Foekens et al., 1994; Van Heerde et al., 2000, 2001, 2002; Andrews et al., 2008).

CHAN4CAST was another well-known forecasting model which was developed by Divakar

et al. (2005). They also employed a regression model capturing the effects of such variables

as past sales, trend, own and competitor prices and promotional variables, and seasonality. In

recent research, Huang, Fildes and Soopramanien (2014) proposed effective methods to

forecast retail SKU sales by incorporating competitive information including prices and

promotions. They found that the proposed methods generate substantially more accurate

forecasts across a range of product categories.

These research studies have made significant contributions to a burgeoning literature on

improving product sales forecasting by integrating more information. However, these studies

all have limitations. First, though models such as SCAN*pro, theoretically considered both

the substitutive and complementary effects, very little past research has empirically

considered the promotional interactive effects in a grocery forecasting system that can work

in practice.. In CHAN4CAST (Divakar et al., 2005), the forecasting system they built is for

consumer packaged goods companies like PepsiCo and Kraft Foods whose goods are sold

through multiple channels in multiple geographic regions. They only empirically considered

the promotional interaction among two beverage brands (Coke and Pepsi). As Cooper et al.

(1999) pointed out “the planning test for retailers is very different from that of manufacturers.

A broad line for a manufacturer may have hundreds of SKUs that could be promoted. This is

small compared to planning for the 30,000 items that are in stock at any given time for a

retailer.” An exception is Huang, Fildes and Soopramanien (2014). Using the weekly data

from a large U.S. retail chain, they included within category competitive (substitutive)

promotional information into their Autoregressive Distributed Lag (ADL) model and

empirically checked the forecasting improvements compared to the model without

competitive information. The key similarities of this study and that by Huang et al. (2014) are

that both studies aim to improve the forecasting accuracy for retailers at SKU level by

integrating extra promotional information from other products. At the same time, there are

some important differences. First, this paper considers both intra- and inter-category

promotional interactions, while Huang et al. (2014) only considered intra-category

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competition. Second, Huang et al. (2014) used a “general to specific” approach to manually

select explanatory variables for every SKU one by one. Though theoretically showing that

integrating intra-category competitive information could improve SKU forecasting accuracy,

the approach is in fact inapplicable: in a real grocery forecasting system, it is impossible to

manually manipulate individual forecasting models for tens thousands of items in a store.

Instead, we propose to use a multi-stage variable selection and model estimation strategy

based on LASSO regression; the total process is fully automatic and therefore can be easily

integrated into a forecasting system. Third, Huang et al. (2014) pooled the SKU sales from 83

stores to simulate a chain level forecasting situation. This does not help a chain manager

allocate SKU stocks at the store level, because of the heterogeneity among stores.

Furthermore, the price and promotional indexes are both aggregated across multiple stores;

this may weaken the explanatory power of these variables. In our research, we focus on store

level sales forecasting, using the raw SKU level information to build a forecasting model

without any aggregation. This is the forecasting situation directly links to a chain or store

manager’s weekly stocking allocation decisions. But this is a more challenging problem, for

the data at the disaggregate level contains more noise than at the aggregate level. Fourth,

Huang et al. (2014) considered 122 SKUs from 6 categories in their empirical study. It is a

large scale empirical study compared to previous existing researches; most of them usually

consider only tens of items in empirical study. This research empirically examines the

forecasts on 926 SKUs in 15 categories for 80 weeks out of sample forecasting. At such a

scale, we need to weigh the complexity of the model and the corresponding computing

efficiency. Therefore, our results will be more realistic, robust and useful in SKU level

decisions.

To summarize, this research is innovative in four respects:

i. The development of a novel fully-automatic algorithm that is capable of selecting key

explanatory variables from a very large data set.

ii. The focus on retail store level modelling and forecasting at SKU level for thousands of

products in order to capture dynamic promotional effects.

iii. The inclusion of both intra- and inter-category information.

iv. The examination of comparative results for a large number of SKUs over a large

number of categories.

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3. Methodology

When cross-category promotional information is considered, the dimensionality of the

promotional explanatory variables grows very rapidly. For example, if the sales of a product

is potentially affected by promotions of items in c categories, each category includes i items,

and each item has j promotion tools, then there are c*i*j potential variables in explanatory

variable space (e.g. for peanut butter in our empirical analysis, the number of variables

considered is 3222). A typical retailer usually has tens thousands of items stocked at any

given time which are usually classified into hundreds of product categories. Obviously it is

unreasonable and infeasible to assume a product is affected by all the products in all the

categories in the store. The method proposed includes four steps which are illustrated in

Figure 1. At step 1, we identify the promotional interactive relationships at category level by

statistical tests. We propose to use a LASSO Granger causality test for such a purpose. At

step 2, we prepare the explanatory variable set for every SKU based on the interactive

categories we identified in the first step. Then we consider three separate approaches dealing

with this high dimensional information: (i) extract only the five top sales products and use

them as the representatives of the category, (ii) preprocess the information to lower the

dimensionality by extracting diffusion factors, and (iii) input all the raw SKU level

promotional information directly into the subsequent LASSO regression. At step 3, to deal

with the high dimensionality remaining in variable space, we propose a three-stage LASSO

strategy to select important predictors and estimate the model parameters: these break down

the variables into predictors from the SKU itself, intra category predictors and finally,

predictors from other categories. At step 4, we generate forecasts for every SKU with the

estimated models.

3.1 Identifying the promotional interactions at category level

To identify which categories are promotional interactive with each other, a simple way is

to resort to expertise by conducting a survey on retailing experts. But the approach is

subjective and subject to the usual biases arising in judgmental decision making. Here we

propose to use a LASSO Granger causality test directly to identity category level promotional

interactions from product sales data.

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Figure 1 Methodology framework

Granger Causality testing is one of the earliest methods developed to quantify the causal

effect from time series observations. It has gained success across many domains due to its

simplicity, robustness, and extendibility (for example, Hiemstra and Jones, 1994). Ashley,

Granger and Schmalensee (1980) gave the definition of causation as follows: Let Ωt,

represent all the information available in the universe at time t. Suppose that at time t

optimum forecasts are made of Yt+1 using all of the information in Ωt, and also using all of

this information apart from the past and present values Xt-j, j>=0, of the series X. If the first

forecast, using all the information, is superior to the second, than the series X has some

special information about Y, not available elsewhere, and X is said to cause Y.

The main challenge in discovering causal relationship among product categories is the

high dimensional time series need to be analyzed in this project. As the number of time series

grows, the statistical significant tests become inefficient, leading to higher chance of spurious

correlations. The LASSO Granger method we considered is one effective way to address this

issue.

Specifically, we first build a set of promotional intensity indexes for every category,

including price indexes, display intensity indexes, feature advertising intensity indexes. All of

these indexes are calculated by weighted averaging the corresponding values across SKUs in

a category. The weight is the weekly average sales of the SKU. That is, the larger the market

share a SKU occupies in a category, the larger the weight it has in the calculation of

promotional intensities. Second, we identify the promotional interactive set for every

category one by one using the LASSO-Granger algorithm (Arnold et al. 2007). In particular,

this can be achieved by solving the following optimization problem:

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2

11 1 2

min ( ) ( )T C

k i i

t i

Y t t

a

a X a (1)

where Xi(t) is the set of promotional intensity indexes in category i at time t; Yk is the average

sales in category k; T is the time length used for the test and C is the total number of

categories considered; a is a coefficient vector to be minimized; λ is a nonnegative penalty

parameter which determines the sparseness of a. The optimal value of λ is determined by

leave-one-out cross-validation in our empirical study. Finally, we determine that the

promotions in category i cause the sales in category k if and only if ai is a non-zero vector.

3.2 Building the explanatory variable space for forecasting

To build a forecasting model for the jth SKU in product category k, three sets of

information make up the potential explanatory variable spaceintra inter{ , , }own

k j kj kj kjS S S . The

information set own

kjS includes all the SKUkj’s own promotional information, its sales history

and time events (e.g., holidays). The information set intra

kjS includes all the promotional

information as well as sales history of SKUs in the category k. Similarly, inter

kjS includes all

the information of SKUs from those identified categories interacting with category k.

Considering the high dimensionality of the potential explanatory variable space, to utilize

the information effectively, we test three approaches in this research. The first is to extract the

information from the five best sale products in the category and use the information to

represent that of the whole category. While the merit of this approach is that it is easy to

implement and less computationally complex, it neglects a large part of the potentially useful

information from other SKUs in the category. The second approach is to perform a Principle

Component Analysis (PCA) on promotional variables to extract a few “factors” as

representative of the whole category sale (Harrell, 2001; Stock and Watson, 2004; Huang et al.

2014). The method utilizes the variance-covariance structure of the predictors with the goal

of finding a few linear combinations of the predictors to explain the covariance structure. In

the empirical study, each explanatory variable, i.e. sales lag, price, display and feature, across

SKUs in the same category is regarded as a cluster. For each cluster, we conduct PCA

dynamically and extract m Principle Components (PCs). So if we have v types of marketing

instrument and c categories, then we extract v*c*m PCs. The PCA is an effective approach to

lower the variable dimensionality, but it has a drawback in forecasting applications.

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Eigen-vectors corresponding to large eigenvalues are retained whereas those associated with

small eigenvalues are discarded. Thus, the retained factors might not have any predictive

power of the dependent variable whereas the discarded factors might be useful (Stock and

Watson, 2002). Here we conduct PCA dynamically as the inputs to the proposed multistage

LASSO. With the aid of LASSO, we can input more diffusion factors into the model (we use

between 270 and 450 factors as candidate predictors in the empirical study). Thus the final

retained factors are no longer only determined by their eigenvalues, but also by their

predictive power. This combines the merit of PCA which is effective in dealing with

collinearity and LASSO which is good at variable selection in high dimensional space while

overcomes the drawbacks of each. The final approach we considered is to input all the raw

information as potential explanatory variables without any preprocessing. Obviously, this

approach keeps all the potential useful information without any loss, but the high

dimensionality in variable space leads to a high computational burden in the steps that follow.

3.3 Variable selection and model estimation with multistage LASSO regression

The main challenge to be faced in developing a SKU-level forecasting model is that the

dimensionality of the promotional explanatory variables space grows very rapidly when

cross-product promotional information is considered, potentially much larger than the length

of the SKU sales’ time series. In order to reduce the high dimensionality effectively and

efficiently, a multistage penalized likelihood method based on the LASSO penalty is applied

to perform the variable selection and parameter estimation simultaneously. LASSO is a

regularization technique for simultaneous estimation and variable selection (Tibshirani, 1996)

which continuously shrinks the coefficients toward 0 as the penalty increases, and some

coefficients are shrunk to exact 0 if the penalty is sufficiently large. Moreover, continuous

shrinkage often improves the prediction accuracy due to the bias variance trade-off.

As noted earlier, important predictors can be highly correlated with some unimportant

ones, but LASSO tends to arbitrarily select only one variable among a group of predictors

with high pairwise correlations. This may results in some unimportant predictors that are

highly correlated with the important predictors being selected by LASSO while important

predictors are missed. In a retailing store, it is very common to promote a set of products

during the same period of time, especially during some special events. This results in the

promotion explanatory variables from different SKUs being highly correlated which makes it

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difficult to distinguish their individual effects on the dependent variable. But from existing

research (Bucklin et al., 1998; Huang et al., 2014), we know that a SKU’s own promotion

explanatory variable are more important than that of other SKUs, and the promotions of

SKUs in the same category as the focus SKU are more important than that of SKUs in other

categories. If we input all the candidate explanatory variables simultaneously into a LASSO

selector, it is likely to select poor variables, i.e., LASSO may select correlated products’

promotion variables instead of the focal SKU’s own predictors. We solve this problem by

proposing a multistage LASSO regression strategy which is illustrated in Figure 2.

Specifically, in order to generate an h weeks ahead forecasting for SKU j in category k,

the variable selection and parameter estimation process is divided into three stages. At the

first stage, only the focal SKU j’s own predictors are inputted into a LASSO regression,

including sales lag, price, display, feature advertising and their lags, calendar events and

week indicators, which can be modeled as an Autoregressive Distributed Lag (ADL) model

(Huang et al., 2014),

2 2

, , , , ,

0 1 1

12 9 1

1 ,

1 1 0

ln( ) ln( ) ln( )L

kj t h kj kjl kj t l kjl kj t h l kjrl kjr t h l kjr kjr t h l

l r r

d c

kjd t h kjc t h v kj t h

d c v

Y Y P D F

W C y

(2)

where

ln (Ykj,t+h) is the log sales of the focal product j in category k in week t+h;

ηkj is the product j’s specific constant;

ln (Pkj,t+h) is the log price of the product j in category k in week t+h;

Dkj1,t+h is an indicator variable for minor display: 1 if product j is minor displayed, in

week t+h; 0 otherwise;

Dkj2,t+h is an indicator variable for major display (including codes lobby and end-aisle in

our empirical data): 1 if product j is major displayed in week t+h; 0 otherwise;

Fkj1,t+h is an indicator variable for minor feature (small and medium size advertisement):

1 if product j is minor featured, in week t+h; 0 otherwise;

Fkj2,t+h is an indicator variable for major feature (large size advertisement and retailer

coupon or rebate): 1 if product j is major featured, in week t+h; 0 otherwise;

d

t hW is the d

th four-week-dummy variable: 1 if t+h is in four-week d of the year; 0

otherwise;

15

c

t h vC is the dummy variable for the c

th calendar event at week t+h-v. When v=0, the

dummy variable represents the week of the calendar event, and the week before the

event if v=1; c take the values from 1 to 9 representing all the calendar events including

Halloween, Thanksgiving, Christmas, New Year’s Day, President’s Day, Easter,

Memorial Day, 4th of July, and Labour Day.

L is the order of lags to be included which is assumed to be one in our empirical study2.

The αkjl is the multiplier for sale lag of product j in category k, the βkjl is the price elasticity,

the γkjrl is the display multiplier, the ρkjrl is the feature multiplier, θdj is the four-week indicator

multiplier, δcj is the calendar multiplier for event c, and the disturbance term is represented by

y1kj,t+h. It is worth noting that the promotional variables are assumed known to the retailer at

t+h in our model, as they usually form part of an agreed promotional plan with suppliers.

We use 10-fold cross-validation to obtain the optimal value of penalty parameter in

LASSO to give minimum cross-validated error. Assuming the data in time window [1, t] is

used for model estimation, after variable selection and parameter estimation by LASSO

regression, we calculate the in-sample forecasts error1 ,1:

ˆkj ty and then generate out-sample

forecasts in this first stage.

At the second stage, we use the in-sample forecasts error,1:

ˆkj ty from the first stage as the

dependent variable, and use the variables from other SKUs in the same category with the

focal SKU as the explanatory variables and model the second stage forecasts by

2 2

1 , , , , 2 ,

1, 1 1

ln( ) ln( )kn

i i i i

kj t h kj kit kj ki t h kjr kir t h kjr kir t h kj t h

i i j r r

y Y P D F y

，(3)

where nk is the number of SKUs (or factors extracted from PCA) in category k and the

disturbance term is represented by 2 ,kj t hy . In the model, if the inputs are factors extracted

from PCA, then the variables Y, P, D and F in the model represent the corresponding factors.

At this stage, variable selection and parameters estimation are again done by LASSO

regression. The in-sample forecasts error2 ,1:

ˆkj ty and out-sample forecasts can then be

calculated for the second stage.

2 The lag structure overcomes many problems of misspecification but taking residual autocorrelation (existent includes

around 15% of SKUs) could lead to further improvements.

16

Figure 2 Multistage LASSO process

17

At the third stage, the in-sample forecasts error 2 ,1:ˆ

kj ty from the second stage are used as

the dependent variable, and the variables from SKUs in the identified influential categories

are used as the explanatory variables,

2 2

2 , , , , , ,

1 1 1

ln( ) ln( )s

k

nsi si si si

kj t h kj si t kj si t h kjr sir t h kjr sir t h kj t h

s S i r r

y Y p D F

(4)

where Sk is the influential category set of category k and the disturbance term is represented

by ,kj t h .We calculate the out-of-sample forecasts for the third stage. The final out-sample

forecasts are the sum of the forecasts in the three stages. We add one-half the mean-squared

error to the forecasts before transforming them back into sales units (Cooper et al. 1999).

4. Empirical study

4.1 Data

The empirical data comes from the IRI dataset (Bronnenberg et al., 2008)3. The IRI

dataset includes grocery and drug chain data from a sample of stores in 50 markets and 30

categories, involving approximately 25%-30% of the consumer packaged goods sales in a

grocery store. This is weekly data by SKU and includes information on sales, price, features

and displays. Based on the objectives of this research, the records from a medium size

grocery store in Chicago as the empirical sample were selected for fifteen product categories

concerned with food and drink. Low-movement SKUs or SKUs which may have been

introduced or discontinued were excluded. Our criterion was that at least 80% of the weeks

must have positive movement a SKU to be included. The empirical dataset includes the

weekly units sold, prices, displays and features of 926 SKUs in 15 food categories for 320

weeks.

Table 1 presents the means and medians of units sold per week and percentages of weeks

concerning promotional activities, including price reductions (more than 5 percent), displays

and features across fifteen categories. It is clear that the price reduction is the most frequent

type of promotion across all the categories. Feature advertising is also frequently used in

many categories, such as frozen pizza and carbonated beverages. Display is only used

3 All estimates and analyses in this paper based on Information Resources, Inc. data are by the author and not by

Information Resources, Inc.

18

occasionally for most of the categories except beer.

As an initial analysis, we can use prior experience to suggest some potential relationships

among categories. For example, substitution might exist in beer and carbonated beverages,

while carbonated beverages and salty snacks, milk and coffee, frozen pizza and beer etc.,

might be complementary. But for some categories, e.g., milk and yogurt, frozen pizza and

coffee, it is difficult to identify by prior experience alone whether an interactive relationship

between them is likely. We therefore resort to the proposed LASSO Granger method to

empirically identify interactions among categories.

Table 1 Description statistics of the data sample

No. Category

Num of SKUs

Mean units sold per week

Median units sold per week

Percentages of weeks concerning promotional activities

Price reductions

Displays Features

1 Beer 98 12.80 7 0.30 0.27 0.13

2 Carbonated

beverages 76 38.25 16 0.42 0.09 0.18

3 Coffee 46 5.90 5 0.34 0.02 0.10

4 Cold cereal 119 15.60 9 0.20 0.05 0.13

5 Frozen dinners 79 18.50 13 0.43 0.04 0.17

6 Frozen pizza 62 21.05 14 0.47 0.10 0.31

7 Frankfurters 21 22.95 10 0.35 0.08 0.16

8 Margarine/Butter 21 29.20 13 0.37 0.05 0.13

9 Mayonnaise 17 15.70 12 0.21 0.03 0.08

10 Milk 40 59.60 24 0.19 0.01 0.06

11 Peanut butter 16 14.30 10 0.22 0.01 0.07

12 Salty snacks 80 17.95 11 0.31 0.12 0.12

13 Soup 129 15.05 9 0.23 0.03 0.10

14 Spaghetti sauce 70 9.40 7 0.38 0.03 0.11

15 Yogurt 52 49.45 37 0.29 0.01 0.08

4.2 Empirical models

We estimate, for each SKU in the sample, ten alternative models which are explained in

19

detail as the following.

(1) ETS. ExponenTial Smoothing (ETS) state space model with seasonality and

non-damped trend (Hyndman et al., 2002).

(2) Base-Lift. A simple exponential smoothing model generates baseline forecasts, and

the promotional adjustment is determined with the increased sales from the most recent

promotion of the focal product. This model is regarded as an industry standard approach, and

has been used as a basic benchmark in a series of recent studies (e.g., Gür Ali et al., 2009;

Huang et al., 2014; Trapero et al. 2014).

(3) ADL-own. ADL Model based on Eq. (2) with only the focal SKU’s own predictors.

(4) ADL-intra-top5. ADL Model based on Eqs. (2) and (3) including the focal SKU’s

own predictors and predictors from the top five sales products in the same category.

(5) ADL-inter-top5. Similar to model (3) but also including the predictors of the top sales

products from identified interactive categories.

(6) ADL-intra-all. ADL Model based on Eqs. (2) and (3) including the focal SKU’s own

predictors and predictors from all the products in the same category

(7) ADL-inter-all. ADL Model with predictors from all the SKUs in both intra- and

inter-categories.

(8) ADL-intra-PCA(x). ADL Model including the focal SKU’s own predictors and x

principle components extracted by PCA from the same category. For example, if x=5, then

for each set of promotional variables in the category we select 5 principle components by

PCA.

(9) ADL-inter-PCA(x). Similar to model (7), but includes both intra- and inter-category

principle components as explanatory variables in the model.

(10) ADL-inter-Ridge. Same as ADL-inter-all, but instead of using LASSO, it is

estimated with ridge regression. Ridge regression is another penalized regression approach, in

which a penalty is applied to the sum of the squared parameter estimates. Unlike LASSO,

ridge regression does not reduce the number of parameters in the model.

4.3 Forecasting evaluation

We use five error measures to compare the forecasting performance of the models. The

first two criteria are Mean Absolute Error (MAE) and Root Mean Square Error (RMSE)

which are traditional and popular scale-dependent error measures. They are easy to calculate,

easy to understand and widely applied. They also have practical meanings to retailing

20

managers, for they naturally place more weight on fast moving SKUs which usually

contribute more revenues than slow moving items in a store. The third criterion is the Mean

Absolute Scaled Error (MASE) which was proposed by Hyndman and Koehler (2006). It can

be considered as a “weighted” arithmetic mean of the MAE based on the variations of the

sales data in the estimation period (Davydenko and Fildes, 2013). It is defined as

11 1

2

t

m

m i i

i

eMASE Mean

Y Y

(5)

where et is the forecast error at week t; m is the number of weeks in the estimation period, Yi

is the sale in week i. MASE is clearly independent of the scale of the data and very suitable

for comparing the forecasts across multiple time series. The drawback of MASE is that it puts

more weights to the data series which are comparatively stable, which makes it vulnerable to

outliers. The fourth criterion we use is based on relative errors. The Average Relative Mean

Absolute Error (AvgRelMAE) is proposed by Davydenko and Fildes (2013) for measuring

forecasting accuracy at SKU-level demand. It is a geometric mean of the ratio of the MAE

between the candidate model and the benchmark model.

1

1

fN Ni

bi i

MAE

MAAvgR lMA

Ee E

(6)

where N is the number of SKUs in the sample, b

iMAE is the MAE of the baseline statistical

forecast for series i, f

iMAE is the MAE of the candidate model f evaluated for series i. The

AvgRelMAE has the advantages of being scale independent and robust to outliers, with a

straightforward interpretation: a value smaller than one indicates an improvement by the

candidate model over the benchmark. In order to measure the forecasting error bias,

potentially important in stocking decisions, the Mean Percentage Error (MPE), that is defined

here as the mean of ratios of total error to total sales per SKU, is used as the last criterion.

The traditional MPE, that is the mean of ratios of error to sales per periods, suffers the

problem of too sensitive to deviations in slow moving periods.

4.4 Forecasting scheme for evaluation

All models are estimated for each SKU separately. We generate the forecasts with both a

fixed forecasting scheme and a rolling scheme. For the fixed scheme, estimation of the

models is based on the data of the first 240 weeks, and the remaining 80 weeks of data are

21

used for forecasting evaluation. Although this is not likely to be used in practice, it helps us to

evaluate a model’s forecast performance over all observations of the validation sample. For

the rolling scheme, we estimate the models with a moving window of 200 weeks and the

forecast for one to four-week ahead horizons. The forecasting horizons are chosen to take into

account typical ordering and planning periods. We move the estimation window forward

week by week throughout the remaining sample period and we re-select variables and

re-estimate the models based on the updated data sets. This differs from Huang, Fildes and

Soopramanien (2014) who used a fixed time window for manually variable selection and

rolling windows for model estimation; here the models in this research are automatically

re-specified for each rolling event based on each new moving time window. Thus, our

forecasting procedure is an iterative one consisting of variable selection, model estimation,

and forecasting throughout the forecasting subsample. It is worth noting that the rolling

scheme is only implemented at SKU level while the identified categories at the first step are

kept fixed.

4.5 Results

4.5.1 Category level interactions

The 240 weeks calibration data is used to analyze the category level interactions. In

Figure 3, a path diagram is presented to represent the Granger relationships among 15

selected product categories.

Every category is represented by a node in the graph and there is a directed line from

category X to Y if and only if X Granger causes Y. Following existing research on consumer

cross category purchasing (Wedel and Zhang, 2004; Walters, 1991; Lee et al., 2013;

Hruschka, 2013), we also find that the interactions between pair of categories are asymmetric

(as shown in Fig.3). For example, Carbonated beverage is affected by the promotion of Salty

snack, but not vice versa. We can also find that some categories are affected by promotions in

many other categories, such as Peanut butter and Spaghetti sauce, while some are more

isolated, such as Soup and Frozen dinner.

22

Figure 3 Promotional interactions at category level

4.5.2 Fixed scheme forecasts

The fixed scheme forecasting results are shown in the left panel of Table 2. The first row

in Table 2 reports the results for the ETS model. This time series model delivers the worst

forecasts among all the empirical models. The Base-Lift performs relatively better than ETS

mainly because it considers promotional information. The ADL-own model is used as a

baseline model to calculate AvgRelMAE which is shown in the third row. All the forecasting

measures for this model are substantially lower than the Base-Lift model which indicates its

better utilization of promotional information. The fourth row reports the results for the

ADL-intra-top5 model which includes extra promotional information from the 5 top sales

SKUs intra-category. The inclusion of extra information does not improve the forecasting

accuracy. The ADL-intra-all model improves the baseline model slightly, while ADL-inter-all

model fails to achieve better forecasts over the baseline. These results indicate that

integrating more information does not necessarily improve the SKU sale forecasts under the

fixed origin scheme. One possible reason is that the extent of promotional interactive effects

among products are time varying and weak at individual SKU level, and the large amount of

extra noisy information increases the risk of overfitting and therefore worsens the forecasts.

23

Table 2 The overall models’ forecasting accuracy with different forecasting scheme and horizons

Fixed scheme Rolling scheme with horizon=1 Rolling scheme with horizon=4

MAE RMSE MASE AvgRelMAE MPE MAE RMSE MASE AvgRelMAE MPE MAE RMSE MASE AvgRelMAE MPE

ETS 8.031 19.105 0.811 1.116 15.8 8.142 19.403 0.822 1.195 16.8 8.148 19.405 0.822 1.195 16.7

Base-Lift 7.752 18.540 0.809 1.088 6.28 7.596 18.155 0.797 1.143 7.03 7.781 18.681 0.822 1.152 7.94

ADL-own* 6.919 14.659 0.770 1.000 -0.39 6.226 13.192 0.710 1.000 1.52 6.368 13.445 0.724 1.000 2.10

ADL-intra-top5 6.885 14.325 0.773 1.005 0.69 6.147 12.971 0.704 0.994 2.97 6.229 13.127 0.713 0.989 3.51

ADL-inter-top5 6.941 14.418 0.778 1.010 2.25 6.125 12.983 0.702 0.991 3.65 6.184 13.079 0.709 0.984 4.13

ADL-intra-all 6.892 14.611 0.770 1.003 1.21 6.041 12.614 0.694 0.980 3.04 6.077 12.652 0.699 0.971 3.27

ADL-inter-all 6.917 14.527 0.775 1.009 2.17 6.023 12.533 0.693 0.979 3.45 6.067 12.581 0.699 0.971 3.73

ADL-intra-PCA(3) 6.875 17.150 0.768 0.997 0.04 6.105 12.949 0.701 0.989 1.71 6.189 13.080 0.710 0.984 2.21

ADL-inter-PCA(3) 6.836 16.738 0.765 0.994 1.20 6.088 13.081 0.699 0.987 2.43 6.165 13.182 0.707 0.980 2.95

ADL-intra-PCA(4) 6.864 16.528 0.769 0.998 -0.19 6.093 12.843 0.701 0.988 1.71 6.169 12.871 0.708 0.983 2.21

ADL-inter-PCA(4) 6.863 18.106 0.766 0.995 0.72 6.071 12.948 0.698 0.985 2.43 6.147 12.888 0.706 0.980 2.95

ADL-intra-PCA(5) 6.838 15.425 0.767 0.997 -0.16 6.103 12.862 0.701 0.989 1.75 6.169 12.974 0.709 0.984 2.05

ADL-inter-PCA(5) 6.805 15.326 0.763 0.993 0.72 6.082 12.856 0.699 0.987 2.47 6.148 12.999 0.707 0.981 2.76

ADL-inter-Ridge 7.716 16.642 0.834 1.107 6.12 6.866 15.277 0.751 1.067 8.10 6.876 15.304 0.752 1.059 8.12

*ADL-own is the benchmark model used to calculate AvgRelMAE; bold text in the table shows the best result in the column

24

In the rows 8 to 13 of the Table 2, however, all the models based on principle

components can improve the forecasts over the baseline model on most of the evaluation

measures. Model ADL-inter-PCA(5) which includesfive principle components for each

promotion variable in a category provides the best forecasts.

The last row in Table 2 reports the results for the ridge regression. Despite utilizing the

full set of information, the forecasting performance of ridge regression is worse than that of

all the models estimated by LASSO. This may because that ridge regression keeps all the

variables in the model without any simplifying selection. So if only a small part of them are

useful, then most of the variables in the model will certainly generate more noise to worsen

the forecasts.

4.5.3 Rolling scheme forecasts

In the middle and right panel of Table 2 we report the results for one week ahead and

four weeks ahead rolling forecasts. In contrast to the results from fixed scheme forecasts, first,

all the models under the rolling scheme deliver substantially better forecasts than that from

the fixed scheme. And more importantly, all the models integrating extra information

estimated by LASSO, even only including extra information of the five top sales products,

perform better than the baseline model. Second, we can see that models combining raw SKU

level information benefit more from the rolling scheme than the models integrating

promotional factors. The ADL-inter-all model outperforms all the other models across all the

accuracy measures. This is an astonishing result compared to its poor performance in the

fixed scheme. The results confirm that the extent of promotional interactions among

individual SKUs are unstable and dynamic across time periods. Third, the factor based

models also perform pretty well though they are not the best. Considering they also perform

well in the fixed schemes, we conclude that they are more robust models than the models

without information pre-extraction. Fourth, by comparing the forecasting improvements

between different horizons, we find the improvements over baseline model become

substantially larger as the forecast horizon increases, e.g. the AvgRelMAE is 0.971 in horizon

4 weeks while it is 0.979 in horizon 1 week for the best performance model. This is in

consistent with the results from Huang et al. (2014). Fifth, Ridge regression with full

information still performs worse than all the models estimated by LASSO in the rolling

scheme. Lastly, the MPEs of LASSO based models are all positive but the values are small,

25

indicating that the overall forecasts from these models are slightly biased towards the

pessimistic. And all the models with extra information were unable to lessen the forecasting

bias compared to the ADL-own model.

Figure 4 shows the MAE improvements of the models with different information sets

over the ETS model. We only compare the MAEs in this figure and the following figures

because the results from four measures are consistent with each other. The incorporation of

the focal product’s own predictors contributes 87.3% of improvements over ETS model. The

extra information from the intra-category five top sales products contributes an additional

5.5%. The following extra information sets contribute less and less. The ADL-intra-all can

only improve over ADL-inter-PCA(4) 1%, and the ADL-inter-all model improves over

ADL-intra-all only 0.6%. All the intra-category information at most contributes about 12%

extra accuracy improvements over the own predictors model, while all the inter-category

information contributes only 0.6% additional improvements.

Figure 4 MAE improvements of the models with different information sets

Table 3 reports the forecasting performance of the various models segmented into

promotion and non-promotion weeks separately. The promotion here is defined as meeting at

least one of three conditions: a temporary price reduction over 5%, major display, or major

feature advertising. All the models’ relative forecasting performance in the two segments is

consistent with the results of the full-sample evaluation reported in Table 2. The

ADL-inter-all model still outperforms all the other models across all accuracy measures in

both segments. This means that the extra information from other SKUs contributes to

26

improving the focal SKU’s forecasting accuracy potentially in both promotion and

non-promotion weeks. We also find that all the models generate better forecasts in

non-promotion weeks than in promotion weeks. Consistent with Gür Ali et al.(2009) and

Huang et al.(2014), time series models have good forecasting performance in non-promotion

weeks; but inconsistent with Huang et al.(2014), ADL-own model outperforms the Base-Lift

in both non-promotion and promotion weeks4.

Table 3 The overall models’ forecasting accuracy in promotion or non-promotion weeks with rolling

scheme and 1-4 week ahead forecasting horizon

No promotion on the focal SKU Promotion on the focal SKU

MAE RMSE MASE AvgRelMAE MPE MAE RMSE MASE AvgRelMAE MPE

ETS 5.70 10.37 1.108 1.246 -19.4 12.85 27.53 0.920 1.225 37.0

Base-Lift 4.907 8.647 0.954 1.195 -7.48 12.37 26.892 0.908 1.106 14.8

ADL-own* 4.411 7.793 0.846 1.000 -2.11 9.296 18.345 0.765 1.000 4.50

ADL-intra-top5 4.335 7.598 0.838 0.994 0.39 9.122 17.986 0.757 0.990 5.41

ADL-inter-top5 4.308 7.563 0.834 0.990 1.05 9.078 17.975 0.752 0.985 6.08

ADL-intra-all 4.267 7.457 0.829 0.985 0.80 8.946 17.487 0.742 0.973 4.91

ADL-inter-all 4.258 7.415 0.828 0.985 1.29 8.922 17.375 0.740 0.972 5.30

ADL-intra-PCA(3) 4.315 7.558 0.835 0.990 -0.19 9.112 18.081 0.753 0.985 3.78

ADL-inter-PCA(3) 4.292 7.526 0.832 0.987 0.55 9.095 18.242 0.748 0.981 4.56

ADL-intra-PCA(4) 4.300 7.540 0.834 0.989 -0.19 9.091 17.889 0.751 0.984 3.93

ADL-inter-PCA(4) 4.277 7.509 0.830 0.986 0.57 9.066 17.984 0.747 0.980 4.71

ADL-intra-PCA(5) 4.301 7.615 0.834 0.989 -0.24 9.091 17.933 0.752 0.986 3.77

ADL-inter-PCA(5) 4.281 7.589 0.830 0.986 0.50 9.072 17.972 0.748 0.983 4.53

ADL-inter-Ridge 4.919 8.457 0.973 1.105 -12.3 10.14 21.488 0.775 1.040 20.3

*ADL-own is the benchmark model used to calculate AvgRelMAE; bold text in the table shows the best result in the column

In Table 4, we compare the forecasting results of three representative models, including

ADL-own, ADL-intra-all and ADL-inter-all, for different categories individually. Those

models are selected because they are the best performing models with the three different

information sets under the rolling scheme. The forecasts are averaged over forecasting

4 In fact, the ADL-own model in this paper is slightly different from the ADL model in Huang et al. (2014). 1) the ‘new’

ADL-own model re-specifies the model for each estimation window, because of its capacity of being automatic, while the ‘old’ ADL-own model used the same model form specified with a pre-set time period (which inevitably to some extent compromised its forecasting performance); 2) the ‘new’ ADL-own model reduces the initial model through LASSO, while the ‘old’ ADL-own relies on a manually implemented general-to-specific modelling strategy. So there might not be a hard evidence of ‘inconsistence’, because this study has actually improved the ADL-own model.

27

Table 4 The models’ forecasting accuracy in various categories with weekly rolling scheme and 1-4 week ahead forecasting horizon

No. Category influential

categories

ADL-own* ADL-intra-all ADL-inter-all

MAE RMSE MASE MPE MAE RMSE MASE AvgRelMAE MPE MAE RMSE MASE AvgRelMAE MPE

1 Beer 2,6,13 4.256 9.297 0.871 4.11 4.050 6.722 0.829 0.959 6.11 4.041 6.711 0.827 0.957 6.47

2 Carbonated

beverages 12 8.530 14.944 0.676 0.07

8.125 14.501 0.657 0.970 2.75

8.044 14.337 0.652 0.963 3.35

3 Coffee 10,12,14 2.138 3.386 0.693 5.07 2.128 3.326 0.690 0.996 6.44 2.135 3.336 0.693 1.000 6.85

4 Cold cereal 2,6,12 5.275 11.412 0.537 -0.66 5.117 11.179 0.527 0.977 2.32 5.129 11.214 0.528 0.979 3.54

5 Frozen dinners -- 6.242 9.586 0.716 0.77 6.136 9.510 0.706 0.990 1.70 6.136 9.510 0.706 0.990 1.70

6 Frozen pizza -- 6.867 11.294 0.768 -4.28 6.724 11.224 0.747 0.977 0.09 6.724 11.224 0.747 0.977 0.09

7 Frankfurters 3,13,15 9.055 23.995 0.438 12.31 9.040 24.512 0.437 0.994 11.32 9.032 23.847 0.440 0.996 9.87

8 Margarine/Butter 13 8.278 23.940 0.708 -6.72 8.074 24.130 0.689 0.978 -6.74 8.038 24.125 0.679 0.969 -4.75

9 Mayonnaise 1,5,6 3.941 6.752 0.721 2.52 3.901 6.736 0.709 0.981 -1.39 3.924 6.791 0.714 0.990 -0.44

10 Milk 1,2,6,12,14 10.185 17.373 1.015 -0.33 8.455 14.884 0.882 0.880 -1.75 8.236 14.204 0.872 0.873 -1.11

11 Peanut butter 4-7,9,10,13,14 4.085 7.712 0.716 -0.63 4.066 7.440 0.729 1.008 2.00 3.996 7.339 0.716 0.991 3.55

12 Salty snacks -- 6.370 11.983 0.788 7.74 6.368 11.948 0.786 0.998 7.54 6.368 11.948 0.786 0.998 7.54

13 Soup -- 5.314 10.701 0.697 5.85 5.223 10.124 0.685 0.980 5.75 5.223 10.124 0.685 0.980 5.75

14 Spaghetti sauce 1,9-13 3.482 5.771 0.657 1.27 3.440 5.565 0.649 0.987 2.47 3.435 5.558 0.649 0.987 3.64

15 Yogurt -- 14.861 27.547 0.753 -1.75 14.133 25.879 0.727 0.965 -2.92 14.133 25.879 0.727 0.965 -2.92

*ADL-own is the benchmark model used to calculate AvgRelMAE

28

Figure 5 MAE improvements of ADL-intra-all and ADL-inter-all over ADL-own

horizon from one to four weeks in the table. In general, both ADL-intra-all and ADL-inter-all

models consistently outperform the baseline model across all categories in terms of

forecasting accuracy. But the extent of the improvements varies among different categories.

Categories such as Cold cereal, Salty snacks and Soup achieve limited forecasting

improvements from extra information. In the category Yogurt and Milk, however, both

models improve the forecasts over the ADL-own model significantly.

In order to show the value of intra- and inter-category information at category level, in

Figure 5, we illustrate the MAE improvements of ADL-intra-all and ADL-inter-all over

ADL-own among different categories. In categories, such as Frankfurters, Margarine/Butter,

Carbonated beverages, Milk and Peanut butter, the contribution from intercategory

information is relatively large, ranging from 12% to 78%, compared with that in other

categories. For Mayonnaise Coffee and Cold cereal, including the intercategory information

in the model could even worsen the forecasts. An explanation is that the useful predictors

from other categories may be too weak to compensate for the loss by including the extra

volume of noisy information from these categories.

In order to investigate whether the proposed LASSO Granger is an effective way to

identify the category level interactive structure, we compare the forecasting results of the

proposed LASSO Granger with the results of a fully connected structure based on

ADL-inter-all under rolling scheme. The full connection means that when forecasting the

29

sales of SKUs in one category, all other 14 categories are considered as influential categories.

The comparison results are illustrated in Figure 6. All the MAEs of LASSO Granger across

categories are smaller or equal to that of using structure of full connections. This means that

connections (Figure 3) identified by LASSO Granger enhance the model’s forecasting

abilities by reducing the redundant noisy data for some categories.

Figure 6 Forecasting comparisons between Full connection and LASSO Granger

Table 5 one week ahead forecasting comparison between one-stage and three-stage LASSO

Model Estimation Scheme MAE RMSE MASE AvgRelMAE MPE

ADL-inter-top5 one-stage fixed 7.024 15.024 0.783 1.013 5.04

ADL-inter-top5 three-stage fixed 6.941 14.418 0.778 1.010 2.25

ADL-inter-all one-stage fixed 7.120 15.036 0.794 1.029 5.68

ADL-inter-all three-stage fixed 6.917 14.527 0.775 1.009 2.17

ADL-inter-PCA(3) one stage fixed 6.937 16.953 0.795 0.998 3.43

ADL-inter-PCA(3) three-stage fixed 6.836 16.738 0.765 0.994 1.20

ADL-inter-top5 one-stage Rolling 6.143 12.967 0.703 0.992 5.13

ADL-inter-top5 three-stage Rolling 6.125 12.953 0.702 0.991 3.65

ADL-inter-all one-stage Rolling 6.118 12.908 0.702 0.992 4.92

ADL-inter-all three-stage Rolling 6.023 12.533 0.693 0.979 3.45

ADL-inter-PCA(3) one stage Rolling 6.193 13.800 0.708 0.995 3.34

ADL-inter-PCA(3) three-stage Rolling 6.088 13.081 0.699 0.987 2.43

30

To show the necessity of the multistage LASSO, we compare the results from both

one-stage and three stage LASSO regression in Table 5. For all models and both fixed and

rolling forecasting schemes, the three stage LASSO methodology produces much more

accurate than the forecasts from the one stage LASSO, and this is true whatever the error

measure.

5. Discussion and Conclusion

In analyzing high-dimensional marketing data, the problem faced is that valuable

predictors of consumer behaviour are often hidden in a large number of useless noisy

variables. When the dimensionality increases with the integration of intra- and inter-

categorical information, the number of unreliable predictors which are correlated with

valuable ones also increases rapidly. This makes the model difficult or even impossible to

estimate. It is also difficult to select the ‘correct’ best specified model because the

corresponding candidate models are many. Various methods have been proposed for selecting

important variables from within the space. A key contribution of this paper is to propose a

novel sequential selection method building on an approach, LASSO, well-known in statistics

but rarely if ever used in marketing where the underperforming stepwise selection method is

most often applied. This new method meets one of the key requirements when analyzing ‘big

data’ of being fully automatic. It is therefore suitable for application in the important

marketing problem of SKU/ store level sales forecasting and promotional planning, when

considering intra- and inter-category promotional information leads to high-dimensionality,

which is this paper’s concern. The second substantive contribution of this paper is that it

develops guidelines to practitioners on whether and how they can improve sales forecasting

accuracy at SKU level by integrating intra- and inter-category promotional information when

they are building a forecasting system for grocery retailers.

Specifically, on the methodological side, we propose a four steps framework to overcome

the high dimensionality of the retail data set that results from integrating the intra- and

inter-category promotional information. Our results show that the scheme of how one

generates the sequence of regression estimates necessary to make forecasts is very important

when integrating extra information. The multi-stage LASSO strategy is the key to improving

the forecasts. This contributes to avoiding the selection of misleading variables among

31

correlated variables by separating different sources of information into several layers. When

considering inter-category information, the first stage in simplifying the problem and

lessening the computational burden is to limit the number of categories to be considered:

LASSO Granger is an effective way to identify the promotional interactions among

categories. Then, various simplification schemes have been evaluated but a key element is to

break down the process of variables selection into three stages: models that include just the

target variables promotional history, those that also include the intra-category variables and

finally, inter-category variables are included. In addition to selecting from amongst these

variable sets, diffusion indices were also developed (based on principal components) that

reduced the dimensionality of these sets. Differing from existing approaches (e.g. Stock and

Watson), we combine diffusion factor with LASSO selection. We first cluster the massive

number of explanatory variables into hundreds of subsets according to their common

attributes (i.e. sales lag, price, display and feature), then for each subset, we conduct PCA

dynamically and extract principle components as the inputs to the proposed multistage

LASSO. This combines the merit of PCA which is effective in dealing with collinearity and

LASSO which is good at variable selection in high dimensional space while make up for their

drawbacks. Finally, a rolling forecasting scheme was shown to effectively utilize extra

information by capturing complex dynamic relationships among products. The total selection

process is fully automatic and therefore can be easily integrated into a forecasting system.

Our substantive results demonstrate which of the methods of variable selection work best

in SKU level retail forecasting. Those models that integrate extra information, even if

including extra information only from the intra-category five top sales products, perform

significantly better than the baseline model when using a rolling forecasting scheme.

Considering various measures of performance, the diffusion approach proved the most robust.

In general, we can improve forecasting accuracy by about12.6% over the baseline model that

includes only the focal SKU’s own predictors. But among the improvements, about 95%

comes from the intra-category information, and only 5% from the inter-category information.

However, the forecasting results at category level show that the accuracy improvements are

spread unevenly among different categories. Though intra-category information still

consistently contributes the main part of the forecasting improvements across categories,

inter-category information can also contribute up to 78% in some categories. But integrating

more information increases the computational complexity substantially: from data processing,

model selection and estimation. In return, better forecasting accuracy can consistently be

32

achieved. In practice, we need to weigh the benefit from increasing forecast accuracy and the

cost and practicality of increasing computational complexity. Because of the rapidly

decreasing cost of data storage, processing and computation, integrating more information to

improve the grocery retailer’s forecasting is a promising option.

When faced with large numbers of potentially explanatory variables it is all too easy for

researchers to identify misleading relationships. In the existing marketing analytics literature,

association-rule discovery or cross category choice models are popular methods to analyze

the correlations between sets of products. These methods are often promoted as a means to

obtain product associations on which to base a retailer’s promotion strategy. Based on this

approach, researchers have argued that associated products with a high lift/interest can be

promoted effectively by only discounting just one of the two products (e.g. Song and

Chintagunta, 2007; Mehta, 2007; Wang & Shao, 2004; Van den Poel et al., 2004). But

Vindevogel et al. (2005) empirically show that this implicit assumption does not hold. A

simple reason is that while associated products are often purchased together, this does not

necessary imply that promotion of one product stimulates the other. The methods proposed in

this paper directly capture this promotional interaction to form a correlation set for every

product to improve their forecasts. They have the advantage of being rigorously validated

through a rolling origin forecasting scheme. Based on the results the methods proposed could

also be used to build a promotional optimization expert system for retailers. This opens a very

interesting direction for further exploration.

Acknowledgments The first author acknowledges the ongoing support of the National Natural Science

Foundation of China under grant nos. 70871057, 71171100, and the support of State Scholarship Fund for

overseas studies. The authors also acknowledge the help from the anonymous referees which has led to

further clarification of the results.

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